To solve the quadratic equation x² + 10x + 25 = 35, we will first simplify the equation by moving all terms to one side.
1. Subtract 35 from both sides:
x² + 10x + 25 – 35 = 0
This simplifies to:
x² + 10x – 10 = 0
2. Next, we can use the quadratic formula to find the values of x. The quadratic formula is:
x = (-b ± √(b² – 4ac)) / 2a
In this equation, a = 1, b = 10, and c = -10.
3. Substitute the values of a, b, and c into the quadratic formula:
x = (−10 ± √((10)² – 4(1)(-10))) / 2(1)
4. Calculate the discriminant:
10² – 4(1)(-10) = 100 + 40 = 140
5. Now substitute the discriminant back into the formula:
x = (−10 ± √(140)) / 2
6. Simplify further:
Simplifying √(140) gives us √(4 × 35), which equals 2√35. So, we have:
x = (−10 ± 2√35) / 2
7. Divide through by 2:
x = -5 ± √35
8. Thus, the final solutions for x are:
x = -5 + √35 and x = -5 – √35. Approximate values for x can be calculated as:
- x ≈ 0.92 (for x = -5 + √35)
- x ≈ -10.92 (for x = -5 – √35)
In conclusion, the solutions to the equation x² + 10x + 25 = 35 are x ≈ 0.92 and x ≈ -10.92.